I am not familiar with the text cited. However you can learn elementary Fourier analysis once you understand elementary calculus. However for a complete understanding of the subject you need to learn measure theory, including Lebesgue integration.

A full-blown rigorous text is outside your capabilities from now if you only know Spivak calculus. In order to really get it, you need at least real analysis (measure theory, topology,...) and functional analysis (= be familiar with Hilbert spaces and stuff).

if you look in spivak's calculus itself you will see a glimpse of the beginning ideas plus a recommendation for further reading on fourier analysis.

in the chapter 15 on trig functions, two problems deal with fourier series, including the riemann lebesgue lemma.

then in the suggested reading he recommends courant volume 1, which has only calculus of one variable as prerequisite obviously at the level of courant, i.e. below spivak, and the little book by robert t. seeley, an introduction to fourier series and integrals, which has some "advanced calculus" prerequisite.

If I recall correctly, there is also a nice little elementary introduction in the differential equations book by edwards and penney.

A full-blown rigorous text is outside your capabilities from now if you only know Spivak calculus. In order to really get it, you need at least real analysis (measure theory, topology,...) and functional analysis (= be familiar with Hilbert spaces and stuff).

if the OP is willing to make the same assumptions that Fourier did (which were objected to by Gauss in his doctoral committee), there is no need to be as rigorous as what you get in functional analysis and normed or Hilbert spaces and such. that assumption is basically: "if the series were to converge to the given [itex]x(t)[/itex] what would the coefficients have to be?" this kind of analysis is done by electrical engineering students all the time right after their 2nd semester of calculus.

If you want to learn Fourier analysis as quickly as possible, I think the only prerequisites that you must have is an excellent grasp of the Riemann integral (Lebesgue is not necessary for a first go, even if you want to learn deep theory + rigor) and also a math major's linear algebra course.

I don't mean to be self promoting, but I started blogging at the same time I joined this forum (i.e. since sunday :P). I am blogging about a couple books I am reading through. One of those books is Fourier Analysis by Elias Stein and Rami Shakarchi.

This is an excellent book (to be expected since Stein is a coauthor) for Fourier analysis with minimal prerequisites (really just the ones I mentioned). Once you have finished this book, you can advanced to the third book of this series (book 2 is complex analysis, which isn't needed for book 3). Book 3 is real analysis, and introduces measure theory, and "fills" in the theory of Book 1.

Edit: When I say as quickly as possible, I mean in terms of preparation. Not in terms of learning the subject quickly.

I echo the recommendation for Stein and Shakarchi's book. It's very well written and has excellent exercises. Spivak's Calculus book should be just about the perfect level of preparation. (Make sure in particular to understand the chapter on uniform convergence.)